# Is the gauge/gravity (or AdS/CFT) duality believed to be exact?

I was wondering about the implications of the gauge/gravity (or AdS/CFT in a more restrictive sense) duality for the way we deal with physical theories, and I was wondering if the duality was believed to be exact, i.e. that the gauge theory and the gravity counterpart are describing exactly the same situation?

If the answer is yes, then as a follow-up question, does that mean that we would not be able to say how many dimensions reality has, in case we were to find an unified gauge theory in $$d$$ dimensions dual with a gravity theory in $$d+1$$ dimensions? We would also then not be able to differentiate if an effect is due to gravity or to gauge theory (for example in the case we were to find a gravity dual to QCD, although this is not a CFT)?

Or is it that the extra degree of freedom induced by the $$+1$$ dimension on the gravity side is somehow encoded in the gauge group?

Edit: Funnily enough, I just came across this statement in the lecture notes by Năstase, which I guess answers the first question (the 2nd one remains):

Re second question: we can formulate quantum mechanics in coordinate representation or equivalently in momentum representation. So do we live in position space or in momentum space? The answer is that it doesn't matter. Some questions are easier to answer in position space, some in momentum space. Everyday life is more easily addressed in position space.

The same is with AdS/CFT: you can describe physics in terms of gravity theory or in terms of a CFT. Some questions are easier to answer in the former formulation, some are easier in the latter. Most experiments are easier to describe in gravity formulation.

Or is it that the extra degree of freedom induced by the +1 dimension on the gravity side is somehow encoded in the gauge group?

I am not sure what you mean by that. The statement of AdS/CFT is that the gravity theory is equivalent to the CFT. This means that you can (in principle) describe the same experiment in gravity theory or in CFT. You don't use these two descriptions simultaneously.

• Hi and thanks for your nice answer! I do get that this is not an isolated case of different formulations for the same theory. The trouble I have, is that it seems that it could be important to know in how many spatial dims we live. But maybe not indeed. What I meant with the last question was, that the extra spatial dimension on the gravity side seems to suggest an extra degree of freedom compared to the gauge side. But if they are equivalent, they must have the same number of degrees of freedom I think. So is a dof redundant in the gravity side, or is there another dof on the gauge side?
– Pxx
Jun 28, 2019 at 18:20
• But I think I can answer that myself, since this is the heart of the holographic principle if I am not mistaken: there is as much information contained in the "surface" theory as in the "volume" dual, so the same number of dof on both sides.
– Pxx
Jun 28, 2019 at 18:24
• As a side note, it seems that in the correspondence between $\mathcal{N}=4$ SYM and IIB superstring theory, the 32 fermionic supercharges of the CFT correspond to the 32 Killing spinors of the gravity, the 6 scalar fields correspond to 6 extra space coordinate, ... thus the correspondence of the degrees of freedom is exact and my 2nd question meaningless as pointed out in the answer.
– Pxx
Jun 28, 2019 at 21:33
• @Jxx you can see how the extra dimension “emerges”, to some degree, by simply looking at the duality between free massive scalar in rigid AdS and generalized free theory on the boundary. However, in this case there is no gravity (and correspondingly the boundary theory is not local), so the entropy behaves not as you described (although is still consistent on both sides). Jun 29, 2019 at 4:19
• You can also think about it this way: the space isometries of AdS map to conformal symmetries. You can see dimensionality of AdS from the number of “translations” that you have in the isometry group. These are interpreted as particular conformal transformations in CFT. Jun 29, 2019 at 4:21